I’m quite busy this week, and so I’ve basically accepted that I am going to fall behind the plan that I laid out in my proposal. I hope to be able to work more quickly in the coming weeks to make up.
I had a meeting on Monday, in which we discussed several exercises from the chapter on cyclic groups. Most of the exercises from this chapter were fairly straightforward once you figure out how to approach them. In other words, they require very few steps to solve, but finding the correct angle of attack can be a challenge. For example, an exercise that we had some trouble with is as follows:
The key here is to observe that the order of every element of a cyclic group divides the order of the group, so if we raise an element to the p^n – 1 power, we should get the identity of the group. Then, if we raise an element to the p^n power, we should get the original element back. If we call the group in question G, then, notationally,
Note that p^n=p*p^(n-1), so we can write
So we have written a as a pth power of an element of the group. Because a is just an arbitrary element of the group, it applies to every element of the group, and so we are done.
This exercise and its solution highlights how the solutions often take advantage of basic divisibility facts, which is why they tend to be fairly straightforward once you find the proper method, but difficult until you manage to find the proper method.